[Theorem 2 of [1]}.]

Pick any distribution \(D\) . Let \(\bar{D}\)  be a corrupted distribution. Suppose that for any \(\mathbf {x}, \mathbf {x}^{\prime } \in \mathcal {X}\) , \(\eta (\mathbf {x}) < \eta (\mathbf {x}^{\prime })\)  implies \(\tilde{\eta }(\mathbf {x}) < \tilde{\eta }(\mathbf {x}^{\prime })\) , where \(\tilde{\eta }\)  is the analogue of \(\eta \)  on noisy labels \(\tilde{y}\)  (i.e. \(\tilde{\eta }(\mathbf {x}) - P(\tilde{Y} = 1 \mid X = \mathbf {x})\) ), and there exists a constant \(C\)  such that \(|\eta (\mathbf {x}) < \eta (\mathbf {x}^{\prime })| \le C \cdot |\tilde{\eta }(\mathbf {x}) < \tilde{\eta }(\mathbf {x}^{\prime })|\) . Then, for any scorer \(s\) ,
\(\text{reg}_{rank}(s; D) \le C \cdot \frac{\bar{\pi }(1 - \bar{\pi })}{\pi (1 - \pi )} \cdot \text{reg}_{rank}(s; \bar{D})\) 
